/*
  This code can be loaded, or copied and pasted, into Magma.
  It will load the data associated to the HMF, including
  the field, level, and Hecke and Atkin-Lehner eigenvalue data.
  At the *bottom* of the file, there is code to recreate the
  Hilbert modular form in Magma, by creating the HMF space
  and cutting out the corresponding Hecke irreducible subspace.
  From there, you can ask for more eigenvalues or modify as desired.
  It is commented out, as this computation may be lengthy.
*/

P<x> := PolynomialRing(Rationals());
g := P![-1, 3, 4, -5, -1, 1];
F<w> := NumberField(g);
ZF := Integers(F);

NN := ideal<ZF | {23, 23, -w^2 + 3}>;

primesArray := [
[3, 3, w^4 - w^3 - 5*w^2 + 3*w + 3],
[9, 3, -w^4 + 5*w^2 - 3],
[9, 3, -w^4 + w^3 + 5*w^2 - 3*w - 2],
[13, 13, -w^4 + w^3 + 4*w^2 - 3*w - 1],
[17, 17, w^4 - w^3 - 5*w^2 + 3*w + 1],
[19, 19, -w^3 + w^2 + 4*w - 2],
[23, 23, -w^2 + 3],
[31, 31, w^3 - 4*w + 2],
[32, 2, 2],
[37, 37, w^3 - 3*w - 1],
[53, 53, -2*w^4 + w^3 + 9*w^2 - 3*w - 2],
[59, 59, -w^4 + 5*w^2 + w - 4],
[61, 61, -w^4 + w^3 + 5*w^2 - 4*w],
[67, 67, -w^4 + 6*w^2 + 2*w - 4],
[71, 71, 2*w^4 - w^3 - 9*w^2 + 4*w + 5],
[79, 79, 2*w^4 - w^3 - 10*w^2 + 2*w + 7],
[83, 83, -w^4 + 2*w^3 + 5*w^2 - 7*w - 2],
[83, 83, -w^4 + w^3 + 4*w^2 - 3*w + 3],
[83, 83, w^4 - w^3 - 5*w^2 + 4*w - 1],
[83, 83, -w^4 + w^3 + 4*w^2 - 4*w - 2],
[83, 83, -2*w^4 + w^3 + 9*w^2 - 3*w - 5],
[97, 97, -w^4 - w^3 + 6*w^2 + 4*w - 4],
[101, 101, -w^4 + 4*w^2 + 2*w - 2],
[107, 107, -2*w^4 + w^3 + 11*w^2 - 3*w - 7],
[127, 127, w^3 - w^2 - 4*w],
[131, 131, w^4 - w^3 - 4*w^2 + 2*w - 1],
[137, 137, w^4 - w^3 - 6*w^2 + 3*w + 3],
[139, 139, -2*w^4 + w^3 + 9*w^2 - 3*w + 1],
[157, 157, 2*w^4 - w^3 - 9*w^2 + 4*w + 2],
[163, 163, -w^2 - 2*w + 3],
[167, 167, w^4 - 5*w^2 + 2*w + 1],
[169, 13, -2*w^4 + 2*w^3 + 9*w^2 - 7*w - 5],
[169, 13, w^4 - 2*w^3 - 5*w^2 + 8*w + 2],
[191, 191, 2*w^4 - w^3 - 9*w^2 + 3*w + 4],
[193, 193, 2*w^4 - 2*w^3 - 8*w^2 + 7*w],
[199, 199, -w^4 + 2*w^3 + 5*w^2 - 9*w - 3],
[211, 211, -3*w^4 + w^3 + 13*w^2 - 2*w - 1],
[227, 227, 2*w^3 - w^2 - 9*w + 1],
[233, 233, -w^3 + w^2 + 4*w + 1],
[251, 251, -3*w^4 + 14*w^2 + w - 2],
[257, 257, -2*w^4 + 2*w^3 + 10*w^2 - 6*w - 5],
[257, 257, -w^4 + w^3 + 5*w^2 - 6*w - 1],
[257, 257, 2*w^4 - 2*w^3 - 11*w^2 + 9*w + 9],
[257, 257, -2*w^4 + 2*w^3 + 9*w^2 - 6*w - 3],
[257, 257, 2*w^4 - w^3 - 10*w^2 + 5*w + 5],
[269, 269, -w^4 + 2*w^3 + 4*w^2 - 6*w - 2],
[271, 271, -w^4 + 4*w^2 + w - 3],
[277, 277, -2*w^4 + 9*w^2 + w - 4],
[281, 281, w^4 - w^3 - 3*w^2 + 3*w - 4],
[283, 283, 2*w^4 - 3*w^3 - 8*w^2 + 13*w],
[289, 17, -w^4 + w^3 + 7*w^2 - 5*w - 5],
[289, 17, 4*w^4 - 3*w^3 - 19*w^2 + 11*w + 6],
[293, 293, -2*w^4 + 11*w^2 - 7],
[317, 317, w^3 - 6*w],
[337, 337, -w^4 + 2*w^3 + 5*w^2 - 9*w - 1],
[337, 337, w^4 - 2*w^3 - 5*w^2 + 6*w + 3],
[337, 337, -w^4 + w^3 + 5*w^2 - 4*w - 6],
[337, 337, -2*w^3 + 8*w + 1],
[337, 337, w^2 + w - 5],
[347, 347, -2*w^4 + w^3 + 9*w^2 - 5*w - 3],
[349, 349, 3*w^4 - 2*w^3 - 13*w^2 + 8*w + 2],
[353, 353, w^4 - 3*w^3 - 4*w^2 + 12*w + 2],
[359, 359, 2*w^3 - w^2 - 7*w + 3],
[361, 19, w^4 - 2*w^3 - 4*w^2 + 7*w + 3],
[361, 19, w^4 - w^3 - 3*w^2 + 3*w - 3],
[367, 367, -2*w^4 + w^3 + 9*w^2 - w - 2],
[379, 379, -2*w^3 + w^2 + 7*w + 2],
[379, 379, -2*w^4 + w^3 + 10*w^2 - 4*w],
[379, 379, 2*w^3 - w^2 - 6*w - 2],
[379, 379, -2*w^3 + 2*w^2 + 6*w - 3],
[379, 379, -3*w^4 + 2*w^3 + 14*w^2 - 7*w - 6],
[383, 383, 2*w^4 + w^3 - 10*w^2 - 7*w + 4],
[383, 383, -2*w^3 + w^2 + 10*w],
[383, 383, w^3 + w^2 - 5*w - 1],
[383, 383, 3*w^4 + 3*w^3 - 15*w^2 - 15*w + 4],
[383, 383, -2*w^4 + 3*w^3 + 10*w^2 - 10*w - 2],
[389, 389, 3*w^4 - w^3 - 14*w^2 + 3*w + 3],
[397, 397, w^4 - 6*w^2 - 2*w + 5],
[397, 397, w^4 - 2*w^3 - 2*w^2 + 6*w - 5],
[397, 397, 2*w^4 - 2*w^3 - 11*w^2 + 10*w + 6],
[397, 397, w^4 - w^3 - 3*w^2 + 4*w - 6],
[397, 397, w^3 + w^2 - 3*w - 4],
[401, 401, w^3 - w^2 - 6*w + 3],
[401, 401, -w^4 + 4*w^2 + 2*w - 3],
[401, 401, -w^4 + 6*w^2 - w - 7],
[421, 421, -w^4 - w^3 + 5*w^2 + 5*w - 4],
[421, 421, -w^4 + w^3 + 6*w^2 - 3*w - 2],
[421, 421, -w^4 - w^3 + 6*w^2 + 6*w - 5],
[421, 421, 3*w^4 - w^3 - 15*w^2 + w + 10],
[421, 421, 2*w^4 - 9*w^2 + 2*w + 4],
[431, 431, 2*w^4 - 11*w^2 - 2*w + 11],
[439, 439, 2*w^4 - 3*w^3 - 9*w^2 + 10*w + 6],
[443, 443, -3*w^4 + 2*w^3 + 14*w^2 - 7*w - 2],
[449, 449, -w^4 + 2*w^3 + 5*w^2 - 6*w - 4],
[461, 461, 3*w^4 - 4*w^3 - 14*w^2 + 14*w + 5],
[463, 463, w^3 - 6*w - 1],
[467, 467, 3*w^4 - 3*w^3 - 13*w^2 + 10*w],
[487, 487, 3*w^4 - w^3 - 14*w^2 + w + 3],
[487, 487, 2*w^2 - w - 4],
[487, 487, -2*w^4 + 2*w^3 + 9*w^2 - 8*w - 5],
[487, 487, 2*w^4 - 10*w^2 - 3*w + 4],
[487, 487, -2*w^4 + w^3 + 8*w^2 - 3*w - 1],
[499, 499, -w^4 + 2*w^3 + 4*w^2 - 8*w - 2],
[499, 499, -2*w^4 + w^3 + 10*w^2 - w - 9],
[499, 499, -w^3 - 2*w^2 + 2*w + 5],
[499, 499, -w^4 + 4*w^2 + 4*w],
[499, 499, -2*w^4 + w^3 + 12*w^2 - 4*w - 9],
[509, 509, 3*w^4 - 2*w^3 - 16*w^2 + 6*w + 9],
[521, 521, -2*w^4 + w^3 + 10*w^2 - 5*w - 4],
[523, 523, w - 4],
[529, 23, -w^3 + 2*w^2 + 4*w - 4],
[529, 23, w^4 + 2*w^3 - 6*w^2 - 11*w + 4],
[569, 569, -2*w^4 + 11*w^2 - 10],
[571, 571, -2*w^4 + w^3 + 10*w^2 - 3*w - 2],
[593, 593, -2*w^4 + 3*w^3 + 10*w^2 - 11*w - 8],
[613, 613, -2*w^3 + 2*w^2 + 9*w - 5],
[617, 617, -2*w^4 + 10*w^2 - w - 7],
[631, 631, -2*w^4 + 2*w^3 + 7*w^2 - 8*w + 3],
[641, 641, -2*w^4 + 2*w^3 + 8*w^2 - 9*w + 3],
[643, 643, -3*w^4 + w^3 + 13*w^2 - w + 1],
[643, 643, w^4 + w^3 - 3*w^2 - 6*w - 2],
[643, 643, -w^4 - w^3 + 6*w^2 + 3*w - 4],
[643, 643, 2*w^4 - w^3 - 10*w^2 + 4*w + 1],
[643, 643, 2*w^4 - 2*w^3 - 8*w^2 + 6*w + 5],
[647, 647, -2*w^3 + w^2 + 6*w - 4],
[659, 659, 4*w^4 - 3*w^3 - 19*w^2 + 11*w + 10],
[661, 661, 2*w^4 + 3*w^3 - 11*w^2 - 14*w + 6],
[673, 673, w^4 - 3*w^3 - 3*w^2 + 10*w - 1],
[683, 683, w^4 - 2*w^3 - 4*w^2 + 6*w - 4],
[701, 701, w^3 - w^2 - 2*w + 4],
[727, 727, -w^4 + 6*w^2 + 2*w - 7],
[743, 743, w^4 - w^3 - 7*w^2 + 5*w + 9],
[769, 769, w^4 + 2*w^3 - 5*w^2 - 9*w + 4],
[787, 787, w^4 + w^3 - 7*w^2 - 3*w + 5],
[797, 797, -2*w^4 - w^3 + 10*w^2 + 4*w - 4],
[797, 797, w^4 - 3*w^3 - 5*w^2 + 12*w + 5],
[797, 797, -3*w^4 + 2*w^3 + 13*w^2 - 8*w],
[797, 797, -w^4 + w^3 + 3*w^2 - 5*w - 1],
[797, 797, 2*w^4 - 11*w^2 + 2*w + 7],
[809, 809, -4*w^4 + 2*w^3 + 18*w^2 - 6*w + 1],
[809, 809, 3*w^4 - 3*w^3 - 14*w^2 + 11*w + 8],
[809, 809, w^4 - 2*w^3 - 2*w^2 + 8*w - 4],
[809, 809, 2*w^4 - 8*w^2 - 2*w + 3],
[809, 809, 2*w^4 - 3*w^3 - 9*w^2 + 13*w + 4],
[821, 821, -2*w^4 + 2*w^3 + 8*w^2 - 6*w + 3],
[823, 823, -2*w^4 + 3*w^3 + 8*w^2 - 12*w - 1],
[829, 829, 2*w^4 - 3*w^3 - 10*w^2 + 13*w + 4],
[839, 839, -2*w^4 + 12*w^2 - w - 10],
[863, 863, 2*w^4 - w^3 - 9*w^2 + 5],
[877, 877, -w^4 + w^3 + 5*w^2 - 5*w - 7],
[881, 881, 2*w^4 - w^3 - 9*w^2 + w - 1],
[887, 887, 3*w^4 - 3*w^3 - 15*w^2 + 11*w + 6],
[907, 907, -3*w^4 + w^3 + 13*w^2 - w - 5],
[919, 919, w^4 + w^3 - 3*w^2 - 5*w - 3],
[929, 929, 2*w^4 - 3*w^3 - 10*w^2 + 9*w + 4],
[937, 937, -2*w^4 + 10*w^2 + 3*w - 7],
[941, 941, -w^4 + 2*w^3 + 3*w^2 - 6*w - 2],
[961, 31, 2*w^4 + 2*w^3 - 10*w^2 - 12*w + 5],
[961, 31, -w^4 + 5*w^2 - w - 7],
[967, 967, -2*w^4 + w^3 + 9*w^2 - 3*w + 2],
[991, 991, -3*w^4 + w^3 + 15*w^2 - 4*w - 5],
[997, 997, -3*w^4 + 15*w^2 - 5],
[997, 997, 2*w^4 - w^3 - 9*w^2 + 6*w + 2],
[997, 997, 2*w^4 - w^3 - 11*w^2 + w + 7],
[997, 997, 2*w^3 - 5*w - 1],
[997, 997, 2*w^4 - 3*w^3 - 6*w^2 + 5*w + 1]];
primes := [ideal<ZF | {F!x : x in I}> : I in primesArray];

heckePol := x^14 + 3*x^13 - 23*x^12 - 66*x^11 + 203*x^10 + 531*x^9 - 906*x^8 - 1928*x^7 + 2296*x^6 + 3044*x^5 - 3349*x^4 - 1252*x^3 + 2214*x^2 - 748*x + 76;
K<e> := NumberField(heckePol);

heckeEigenvaluesArray := [e, 87265/2753488*e^13 + 104137/1376744*e^12 - 2271245/2753488*e^11 - 4898289/2753488*e^10 + 5798719/688372*e^9 + 43131035/2753488*e^8 - 119118121/2753488*e^7 - 176657031/2753488*e^6 + 324775019/2753488*e^5 + 331566585/2753488*e^4 - 223777233/1376744*e^3 - 106403649/1376744*e^2 + 31023785/344186*e - 8902813/688372, -76753/2753488*e^13 - 27161/1376744*e^12 + 2152181/2753488*e^11 + 1076593/2753488*e^10 - 5798209/688372*e^9 - 7318763/2753488*e^8 + 120488913/2753488*e^7 + 19111383/2753488*e^6 - 308639635/2753488*e^5 - 10029761/2753488*e^4 + 174555245/1376744*e^3 - 9288727/1376744*e^2 - 14473387/344186*e + 3818017/688372, 281475/2753488*e^13 + 109219/344186*e^12 - 6126669/2753488*e^11 - 18866695/2753488*e^10 + 24395563/1376744*e^9 + 147833247/2753488*e^8 - 179727811/2753488*e^7 - 518457295/2753488*e^6 + 329158197/2753488*e^5 + 795529109/2753488*e^4 - 161735311/1376744*e^3 - 197623161/1376744*e^2 + 11857449/172093*e - 3266217/688372, 38443/688372*e^13 + 345187/1376744*e^12 - 1420917/1376744*e^11 - 931768/172093*e^10 + 8484411/1376744*e^9 + 58460721/1376744*e^8 - 8969817/688372*e^7 - 206378121/1376744*e^6 + 6938649/688372*e^5 + 325105455/1376744*e^4 - 5170413/172093*e^3 - 86871895/688372*e^2 + 19949243/344186*e - 2158867/344186, -158225/1376744*e^13 - 167459/688372*e^12 + 3953511/1376744*e^11 + 7368029/1376744*e^10 - 4785067/172093*e^9 - 59583401/1376744*e^8 + 184512777/1376744*e^7 + 219633899/1376744*e^6 - 471084497/1376744*e^5 - 359808495/1376744*e^4 + 305606371/688372*e^3 + 87210007/688372*e^2 - 40271091/172093*e + 16438763/344186, -1, 96751/2753488*e^13 + 95051/688372*e^12 - 2131157/2753488*e^11 - 8691163/2753488*e^10 + 9081793/1376744*e^9 + 74038183/2753488*e^8 - 82795367/2753488*e^7 - 293360743/2753488*e^6 + 242956393/2753488*e^5 + 534662693/2753488*e^4 - 227346507/1376744*e^3 - 160003693/1376744*e^2 + 23389309/172093*e - 22066501/688372, 891529/2753488*e^13 + 704929/688372*e^12 - 19730775/2753488*e^11 - 61640253/2753488*e^10 + 81538163/1376744*e^9 + 491954421/2753488*e^8 - 654146225/2753488*e^7 - 1774198225/2753488*e^6 + 1439901715/2753488*e^5 + 2838835127/2753488*e^4 - 945735705/1376744*e^3 - 729805447/1376744*e^2 + 80819728/172093*e - 55145747/688372, -1748999/2753488*e^13 - 383663/172093*e^12 + 37183937/2753488*e^11 + 134245451/2753488*e^10 - 144527643/1376744*e^9 - 1072243003/2753488*e^8 + 1062828471/2753488*e^7 + 3876690315/2753488*e^6 - 2149333985/2753488*e^5 - 6258353753/2753488*e^4 + 1433408091/1376744*e^3 + 1674075965/1376744*e^2 - 140880980/172093*e + 84630349/688372, -578021/1376744*e^13 - 1973201/1376744*e^12 + 1519657/172093*e^11 + 42619771/1376744*e^10 - 92198823/1376744*e^9 - 167173495/688372*e^8 + 319775109/1376744*e^7 + 588651949/688372*e^6 - 571975543/1376744*e^5 - 913115517/688372*e^4 + 332556433/688372*e^3 + 228873969/344186*e^2 - 134212665/344186*e + 9610605/172093, 1309775/2753488*e^13 + 1189305/688372*e^12 - 27022709/2753488*e^11 - 103518147/2753488*e^10 + 99099621/1376744*e^9 + 821386343/2753488*e^8 - 648292991/2753488*e^7 - 2946920831/2753488*e^6 + 1074530753/2753488*e^5 + 4733482605/2753488*e^4 - 647940079/1376744*e^3 - 1299359941/1376744*e^2 + 76993625/172093*e - 26126653/688372, -413633/1376744*e^13 - 202769/172093*e^12 + 8208985/1376744*e^11 + 35286601/1376744*e^10 - 27970083/688372*e^9 - 279813251/1376744*e^8 + 156915057/1376744*e^7 + 1003617033/1376744*e^6 - 194264057/1376744*e^5 - 1617304373/1376744*e^4 + 118687297/688372*e^3 + 453570527/688372*e^2 - 43076594/172093*e + 3954145/344186, -145505/2753488*e^13 - 260721/688372*e^12 + 1404851/2753488*e^11 + 22476933/2753488*e^10 + 5823881/1376744*e^9 - 175630609/2753488*e^8 - 181180263/2753488*e^7 + 616371097/2753488*e^6 + 706858545/2753488*e^5 - 969800915/2753488*e^4 - 483809539/1376744*e^3 + 284428075/1376744*e^2 + 17938019/172093*e - 22501325/688372, 212915/688372*e^13 + 1827761/1376744*e^12 - 7743237/1376744*e^11 - 9810713/344186*e^10 + 42095333/1376744*e^9 + 304779829/1376744*e^8 - 18914839/688372*e^7 - 1057339323/1376744*e^6 - 61532141/344186*e^5 + 1618016707/1376744*e^4 + 118268751/344186*e^3 - 428604137/688372*e^2 - 9713265/344186*e + 14051409/344186, 1659071/1376744*e^13 + 5960223/1376744*e^12 - 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286422043/688372, 2042561/1376744*e^13 + 3679557/688372*e^12 - 43293097/1376744*e^11 - 161464581/1376744*e^10 + 84078811/344186*e^9 + 1296191467/1376744*e^8 - 1248501513/1376744*e^7 - 4727862555/1376744*e^6 + 2614314703/1376744*e^5 + 7772893257/1376744*e^4 - 1824833403/688372*e^3 - 2184995323/688372*e^2 + 357330480/172093*e - 90696811/344186, 163349/1376744*e^13 + 1441733/1376744*e^12 - 509593/344186*e^11 - 33209179/1376744*e^10 - 462969/1376744*e^9 + 141833477/688372*e^8 + 75717227/1376744*e^7 - 558303573/688372*e^6 - 200420153/1376744*e^5 + 1000570039/688372*e^4 - 85259439/688372*e^3 - 152121195/172093*e^2 + 159359645/344186*e - 16931735/172093, 3597717/1376744*e^13 + 12400445/1376744*e^12 - 38862363/688372*e^11 - 272715879/1376744*e^10 + 622655255/1376744*e^9 + 274878759/172093*e^8 - 2424571365/1376744*e^7 - 4040850873/688372*e^6 + 5369366657/1376744*e^5 + 3354852277/344186*e^4 - 3824640281/688372*e^3 - 1880049947/344186*e^2 + 1449389369/344186*e - 114771414/172093, -163292/172093*e^13 - 2883589/688372*e^12 + 2941804/172093*e^11 + 30983119/344186*e^10 - 63017711/688372*e^9 - 120460619/172093*e^8 + 13665016/172093*e^7 + 1674513181/688372*e^6 + 349255469/688372*e^5 - 1284143181/344186*e^4 - 304493769/344186*e^3 + 339451925/172093*e^2 - 10341411/172093*e - 15798852/172093, 4260513/2753488*e^13 + 4283873/688372*e^12 - 84229963/2753488*e^11 - 374283717/2753488*e^10 + 285602743/1376744*e^9 + 2984080033/2753488*e^8 - 1598890817/2753488*e^7 - 10770967697/2753488*e^6 + 2052140535/2753488*e^5 + 17426405163/2753488*e^4 - 1418511789/1376744*e^3 - 4800612111/1376744*e^2 + 261662560/172093*e - 114684883/688372, -5585101/1376744*e^13 - 4970927/344186*e^12 + 117301867/1376744*e^11 + 433949493/1376744*e^10 - 445263053/688372*e^9 - 3455306645/1376744*e^8 + 3125673525/1376744*e^7 + 12435636365/1376744*e^6 - 5873018183/1376744*e^5 - 19957236255/1376744*e^4 + 3779943721/688372*e^3 + 5338285375/688372*e^2 - 794039009/172093*e + 214934123/344186, 10181821/2753488*e^13 + 2295035/172093*e^12 - 211724995/2753488*e^11 - 799729817/2753488*e^10 + 789159561/1376744*e^9 + 6350699833/2753488*e^8 - 5350227893/2753488*e^7 - 22791219105/2753488*e^6 + 9500640723/2753488*e^5 + 36549229467/2753488*e^4 - 5962598801/1376744*e^3 - 9908188235/1376744*e^2 + 666534525/172093*e - 311256299/688372, 2238871/2753488*e^13 + 4198195/1376744*e^12 - 44098207/2753488*e^11 - 179179399/2753488*e^10 + 73589287/688372*e^9 + 1379942577/2753488*e^8 - 768627215/2753488*e^7 - 4727628161/2753488*e^6 + 682646905/2753488*e^5 + 7056330851/2753488*e^4 - 247493255/1376744*e^3 - 1679680511/1376744*e^2 + 163160807/344186*e - 55856011/688372, 461521/2753488*e^13 + 78293/172093*e^12 - 10701775/2753488*e^11 - 27379757/2753488*e^10 + 47822789/1376744*e^9 + 222564277/2753488*e^8 - 435962289/2753488*e^7 - 854664637/2753488*e^6 + 1143799551/2753488*e^5 + 1584835895/2753488*e^4 - 870519693/1376744*e^3 - 536001747/1376744*e^2 + 73787845/172093*e - 55915111/688372, -3456699/2753488*e^13 - 2919535/688372*e^12 + 72680913/2753488*e^11 + 252654199/2753488*e^10 - 272602449/1376744*e^9 - 1984126003/2753488*e^8 + 1802707723/2753488*e^7 + 6980808067/2753488*e^6 - 2756060421/2753488*e^5 - 10797786785/2753488*e^4 + 1222698931/1376744*e^3 + 2769550473/1376744*e^2 - 132966490/172093*e + 17292217/688372, -1472237/2753488*e^13 - 2913685/1376744*e^12 + 30076913/2753488*e^11 + 128048781/2753488*e^10 - 55270355/688372*e^9 - 1029690327/2753488*e^8 + 768873653/2753488*e^7 + 3761389179/2753488*e^6 - 1618920119/2753488*e^5 - 6173827293/2753488*e^4 + 1339548781/1376744*e^3 + 1678611533/1376744*e^2 - 308502285/344186*e + 125441845/688372, 3730593/1376744*e^13 + 6504897/688372*e^12 - 77638473/1376744*e^11 - 281442297/1376744*e^10 + 143967347/344186*e^9 + 2212490495/1376744*e^8 - 1898796401/1376744*e^7 - 7816124927/1376744*e^6 + 3044274103/1376744*e^5 + 12221594173/1376744*e^4 - 1581211597/688372*e^3 - 3190729037/688372*e^2 + 349117073/172093*e - 57478829/344186, 5979781/2753488*e^13 + 10220713/1376744*e^12 - 127727093/2753488*e^11 - 446188005/2753488*e^10 + 249307409/688372*e^9 + 3554811611/2753488*e^8 - 3664807405/2753488*e^7 - 12808487459/2753488*e^6 + 7291041667/2753488*e^5 + 20579491537/2753488*e^4 - 4713897125/1376744*e^3 - 5468452213/1376744*e^2 + 921875865/344186*e - 277214889/688372, -9126917/2753488*e^13 - 4002891/344186*e^12 + 193710167/2753488*e^11 + 700587729/2753488*e^10 - 749218065/1376744*e^9 - 5600015661/2753488*e^8 + 5439785325/2753488*e^7 + 20278659313/2753488*e^6 - 10734403439/2753488*e^5 - 32850558239/2753488*e^4 + 7062442869/1376744*e^3 + 8881325455/1376744*e^2 - 717493775/172093*e + 416579759/688372, -1716631/1376744*e^13 - 7385759/1376744*e^12 + 4024691/172093*e^11 + 160606045/1376744*e^10 - 192397961/1376744*e^9 - 635563491/688372*e^8 + 347142151/1376744*e^7 + 2267545063/688372*e^6 + 218574495/1376744*e^5 - 3612130633/688372*e^4 - 235737855/688372*e^3 + 498494018/172093*e^2 - 209586545/344186*e - 4172483/172093, -1037741/1376744*e^13 - 2274873/1376744*e^12 + 12663847/688372*e^11 + 48674483/1376744*e^10 - 239411051/1376744*e^9 - 94605433/344186*e^8 + 1132088053/1376744*e^7 + 661902085/688372*e^6 - 2863485229/1376744*e^5 - 254794395/172093*e^4 + 1843641151/688372*e^3 + 114588521/172093*e^2 - 468231969/344186*e + 44111871/172093, -774803/688372*e^13 - 1100727/344186*e^12 + 4291381/172093*e^11 + 46815233/688372*e^10 - 70331755/344186*e^9 - 89666043/172093*e^8 + 543414119/688372*e^7 + 1215768507/688372*e^6 - 535670949/344186*e^5 - 882347397/344186*e^4 + 560741541/344186*e^3 + 381639581/344186*e^2 - 149519966/172093*e + 21930901/172093, -9846013/2753488*e^13 - 17483107/1376744*e^12 + 206738037/2753488*e^11 + 762314709/2753488*e^10 - 98226847/172093*e^9 - 6066724251/2753488*e^8 + 5556172165/2753488*e^7 + 21864195231/2753488*e^6 - 10670657263/2753488*e^5 - 35307087513/2753488*e^4 + 7063187905/1376744*e^3 + 9587737605/1376744*e^2 - 1479488739/344186*e + 419808985/688372, -1117821/2753488*e^13 - 239581/344186*e^12 + 28911983/2753488*e^11 + 41954425/2753488*e^10 - 143872717/1376744*e^9 - 335812165/2753488*e^8 + 1390684661/2753488*e^7 + 1211686689/2753488*e^6 - 3365977191/2753488*e^5 - 1914389127/2753488*e^4 + 1868074321/1376744*e^3 + 483934211/1376744*e^2 - 88671629/172093*e + 26069799/688372, -2094563/688372*e^13 - 7906175/688372*e^12 + 10576832/172093*e^11 + 171421361/688372*e^10 - 298912789/688372*e^9 - 338103222/172093*e^8 + 911766173/688372*e^7 + 1202297018/172093*e^6 - 1354523133/688372*e^5 - 3811626609/344186*e^4 + 826440257/344186*e^3 + 2047474807/344186*e^2 - 467161007/172093*e + 48624988/172093];
heckeEigenvalues := AssociativeArray();
for i := 1 to #heckeEigenvaluesArray do
  heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i];
end for;

ALEigenvalues := AssociativeArray();
ALEigenvalues[ideal<ZF | {23, 23, -w^2 + 3}>] := 1;

// EXAMPLE:
// pp := Factorization(2*ZF)[1][1];
// heckeEigenvalues[pp];

print "To reconstruct the Hilbert newform f, type
  f, iso := Explode(make_newform());";

function make_newform();
 M := HilbertCuspForms(F, NN);
 S := NewSubspace(M);
 // SetVerbose("ModFrmHil", 1);
 NFD := NewformDecomposition(S);
 newforms := [* Eigenform(U) : U in NFD *];

 if #newforms eq 0 then;
  print "No Hilbert newforms at this level";
  return 0;
 end if;

 print "Testing ", #newforms, " possible newforms";
 newforms := [* f: f in newforms | IsIsomorphic(BaseField(f), K) *];
 print #newforms, " newforms have the correct Hecke field";

 if #newforms eq 0 then;
  print "No Hilbert newform found with the correct Hecke field";
  return 0;
 end if;

 autos := Automorphisms(K);
 xnewforms := [* *];
 for f in newforms do;
  if K eq RationalField() then;
   Append(~xnewforms, [* f, autos[1] *]);
  else;
   flag, iso := IsIsomorphic(K,BaseField(f));
   for a in autos do;
    Append(~xnewforms, [* f, a*iso *]);
   end for;
  end if;
 end for;
 newforms := xnewforms;

 for P in primes do;
  xnewforms := [* *];
  for f_iso in newforms do;
   f, iso := Explode(f_iso);
   if HeckeEigenvalue(f,P) eq iso(heckeEigenvalues[P]) then;
    Append(~xnewforms, f_iso);
   end if;
  end for;
  newforms := xnewforms;
  if #newforms eq 0 then;
   print "No Hilbert newform found which matches the Hecke eigenvalues";
   return 0;
  else if #newforms eq 1 then;
   print "success: unique match";
   return newforms[1];
  end if;
  end if;
 end for;
 print #newforms, "Hilbert newforms found which match the Hecke eigenvalues";
 return newforms[1];

end function;